Input third-degree price discrimination by congestible facilities

Transcription

1 Input third-degree price discrimination by congestible facilities Hugo E. Silva Department of Spatial Economics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Tinbergen Institute, Gustav Mahlerplein 117, 1082 MS, Amsterdam, The Netherlands. Preliminary draft. October, Abstract This paper studies third-degree price discrimination by transport facilities, such as airports and seaports, which sell access to the infrastructure as a necessary input for downstream production. These facilities are prone to congestion which makes downstream markets interrelated and their ownership structure is diverse, varying from public (domestic welfare maximizing) to private (profit maximizing). We show that input price discrimination by a private supplier can increase aggregate output and increase welfare in a setting where, in absence of congestion, output does not change and welfare is reduced when price discrimination is allowed. Therefore, the presence of negative consumption externalities enlarges the extent to which input price discrimination is desirable. We also analyze the effects of price discrimination by a public facility and describe the conditions under which banning input price discrimination is efficient for both types of ownership forms. We argue that there is a limited scope for this to occur, which suggests that the current practice of enforcing a broad ban on input price discrimination that covers congestible facilities with different ownership forms may have to be revised. 1. Introduction Congestible facilities often provide infrastructure access to downstream firms that may operate in different markets. For example, access to the airport s runway is an essential input for an airline s production in multiple city-pairs. In many countries, input price discrimination is banned by law, so that suppliers must charge uniform prices to firms. To a large extent the ban on input price discrimination applies also to congestible facilities. For example, the EU Airport Charges directive (2009/12/EC) prohibits differentiated charges to airlines using the same service (i.e. terminal and level of service). 1 A similar ban holds for airports in the U.K. (Section 41 of the 1986 Airports Act) and in the U.S. (2013 FAA s Policy Financial support from ERC Advanced Grant # (OPTION) is gratefully acknowledged. I am grateful to Vincent van den Berg, Achim Czerny and to Erik Verhoef for their helpful comments. address: h.silvamontalva@vu.nl (Hugo E. Silva) 1 The ban applies to the airport with the highest passenger movement in each EU Member State and to any airport whose annual traffic is over 5 million passengers.

2 Regarding Airport Rates and Charges). 2 The regulations of the World Trade Organization (WTO) through the General Agreement on Tariffs and Trade (GATT) basically do not allow price discrimination by ports. Similar examples can be found in other transport sectors too. It is therefore evident that current bans on price discrimination to congestible facilities have an impact on many large economic sectors. Congestible facilities feature two characteristics that make the analysis distinct from the traditional price discrimination studies in input markets. First, there is congestion: an output increase by one firm imposes additional costs on consumers of all markets, therefore reducing the price that other firms can charge. Congestion, thus, makes demands interrelated in a way analogous to substitution. In addition, downstream firms do not fully internalize this externality, so that the aggregate output may be inefficiently high. Second, the ownership form of the congestible facilities subject to price discrimination regulation is diverse: for example, in Europe alone, the ban applies to private, public and mixed private-public airports. The incentives of the facilities to apply price discrimination and, therefore, its effect on welfare may vary with the ownership form. The purpose of this paper is to study third-degree price discrimination by both private (profit maximizing) and public (domestic welfare maximizing) congestible facilities, and shed light on whether and when a broad (e.g. Europe-wide) ban on input price discrimination is desirable. In order to focus on the effects of congestion and ownership form, we analyze a case with two downstream markets and two downstream firms, one domestic and one foreign, that are equally efficient. Each firm is a monopoly in one market and the only interdependency is through congestion. To highlight the differences with the uncongested case, we consider an industry structure that is comparable to the commonly used structure in the literature in that the input provider is a monopolist and firms take the input price as given. A private facility would therefore differentiate charges according to the different demand conditions in the two markets. A public facility also considers domestic consumer surplus and the profit of the domestic firm, so it would give price concessions to the domestic firm in detriment of the foreign firm, to stimulate (domestic) production and capture foreign profit. In this paper we, first, analyze the price, output and welfare effect of input thirddegree price discrimination by a private facility and assess how the presence of congestion externalities affects the analysis. Second, we study the case of a public facility and assess the impact of the ownership form on the effects of price discrimination. Finally, we compare the welfare effect of price discrimination under both ownership forms and elaborate on the desirability of a broad ban on price discrimination. Input third-degree price discrimination when downstream firms are equally efficient and operate in multiple markets has been recently explored by Arya and Mittendorf (2010). In 2 The FAA s Policy Regarding Airport Rates and Charges prohibits unjust discrimination. This prohibition does not prevent airports to set different charges to different aeronautical users (such as signatory and nonsignatory carriers) or to in peak and off-peak periods. Nevertheless, it explicitly bans, for example, differentiated charges to firms that belong to the same category irrespective of the markets they serve, and to foreign and domestic airlines engaged in similar international air services. 2

3 a two-market setting and using linear demands, they show that the aggregate output is the same under both pricing regimes (i.e. with and without discrimination), but price discrimination leads to an output shift from the market with higher demand to the market with lower demand. Therefore, extending the intuition from final good markets, price discrimination leads to welfare deterioration in the case where there is only one firm operating in each market. 3,4 Our paper contributes to this branch of the literature by studying the case with interrelated demands through congestion and by considering a domestic-welfaremaximizing input provider. Also with linear demands, yet in the presence of congestion, we find benefits from price discrimination when the willingness to pay to reduce travel delays differs across markets. This may be due to different average income of consumers or different composition of trip purpose (leisure versus business) across markets, among others. We show that under private and public ownership, input price discrimination can increase aggregate output and welfare. This result suggests that the presence of congestion externalities enlarges the extent to which input price discrimination is desirable. The literature on price discrimination under negative consumption externalities has mainly focused on final markets. Adachi (2005), considering only consumption externalities within markets, shows that welfare can increase when third-degree price discrimination is allowed when output does not. Czerny and Zhang (2015) study third-degree price discrimination by a monopoly airline considering cross- and within-market negative externalities together, a feature that is typical of congestion. They show that there is a time-valuation effect of price discrimination that works in the opposite direction as the output effect and, as a result, welfare can increase when output decreases. When demands are linear, they find that price discrimination reduces the aggregate passenger quantity, which reduces congestion costs, and that this can increase welfare. We show that price discrimination in input markets under congestion externalities exhibits fundamental differences with the case of final markets and that the analysis provides essentially different insights. We find that under linear demands, input third-degree price discrimination by a profit maximizing facility can yield higher total welfare and consumer surplus than what is obtained under uniform pricing by leading to aggregate output expansion and to a reduction of both prices. As congestion effects work in a similar way as the substitution effect, the intuition is similar to the one provided by Layson (1998) for substitute final goods. Reducing the price in one market can reduce the profitability of the other and, if this effect is large enough, it can cause that the price in the other market has to also be reduced. These results are in sharp 3 They also analyze the case in which a firm operates in both markets and faces different degrees of competition in each one. When the market with lower demand is also the market with lower competition, the increased production incentives under price discrimination in this market may increase welfare. 4 There is also a large stream of literature studying third-degree price discrimination in input markets where downstream firms have different levels of (cost) efficiency that shows benefits of uniform pricing (e.g., Katz, 1987; DeGraba, 1990; Yoshida, 2000; Valletti, 2003). Nevertheless, uniform pricing can be harmful when there is bargaining between buyers and suppliers (O Brien and Shaffer, 1994), and when there is input demand-side substitution (Inderst and Valletti, 2009). 3

4 contrast with the outcome of the models in final markets. The difference arises because the input provider faces derived demands, which may have essential differences with final good demands. For example, under price discrimination by a private supplier, the market with the highest input price can be the market with the lowest final good price. Therefore, what could be called the weak market in terms of final price can be the strong market for the input provider. Our analysis also contributes to the transport policy literature. Benoot et al. (2013) study price discrimination by a local welfare maximizing airport when passengers are homogenous and airlines and markets are symmetric. The incentives for price discrimination arise from that the foreign passengers surplus is not fully considered. They numerically find that welfare is higher under uniform pricing because foreign passengers surplus increases. 5 We focus on the more fundamental question of whether and when a broad ban on price discrimination is welfare enhancing. We find that under uniform pricing total welfare may be higher than under price discrimination by a domestic welfare maximizing facility also when there is asymmetry, but this is not the only possible outcome as price discrimination can increase welfare. Importantly, we find that under the conditions that make uniform pricing by a public facility welfare enhancing, price discrimination may yield higher total welfare than uniform pricing if the input supplier is private. We also find that the reverse may happen: price discrimination by a public facility can increase total welfare under the same conditions that make uniform pricing socially optimal if the supplier is private. These results have important policy implications. The ownership form of transport facilities has been consistently moving from public to private in the last decades; for example, in 2010, 48% of all European traffic was handled by a fully privatized airport or by mixed privatepublic airports. Our results suggests that a ban on price discrimination that covers a large number of congestible facilities and, in particular, that covers different ownership forms has to be revised, especially in the light of the privatization wave. The remainder of the paper is structured as follows. Section 2 introduces the model and main assumptions. Section 3 analyzes the effects of price discrimination by a private facility while Section 4 analyzes the case of a public facility. Section 5 compares the welfare effect of price discrimination under both ownership forms. Section 6 extends the conclusions to downstream perfect price discrimination and Section 7 concludes. 2. The model and the downstream markets We study price discrimination by a monopolist transport facility that sells access to its infrastructure, which is an input necessary for downstream production. There are two downstream markets served by the facility, A and B, which may represent movement of 5 Haskel et al. (2013) study price discrimination by substitute private airports when airlines and markets are symmetric. In their model, the incentives to price discriminate arise from that there is bargaining between airports and airlines, so that the possibility of differentiated prices changes the bargaining structure. Their main result is that price discrimination leads to lower prices as it makes airlines tougher negotiators. 4

5 people or cargo to different destinations. Markets are interrelated through congestion as an additional unit of output in any market imposes an externality on all other consumers, but are otherwise independent. Downstream firms transform one unit of input into one unit of output. Think, for example, of an airport setting per-passenger charges to airlines flying to different cities, where congestion occurs at the passenger s facilities (security passenger and baggage screening or access to gateways) and/or on the runway as a result of aircraft landing and take-off. There are two downstream firms and we denote them in the same way as the markets in which they operate. Thus, firm i operates in market i with i = {A, B}. Each firm s demand q i depends on the full price faced by consumers, which is the sum of the downstream firm s price (e.g. ticket) and the cost of congestion (e.g. delays at the airport). The delay due to congestion, D(Q), increases in the aggregate consumption (Q = q A + q B ), to reflect withinand cross-market negative consumption externalities. As every unit of output causes delays on all others, a natural interpretation for the market demand is that it is the aggregation of consumers who buy either 0 or 1 unit of the good (e.g. a trip in a peak period) and are heterogeneous in their willingness to pay for the good. Denote P i (q i ) the downward-sloping inverse demand in market i and v i the willingness to pay to reduce congestion delays, or the value of time as shorthand, which is assumed to be the same for all individuals in a market but to be different across markets. Without loss of generality we assume that consumers in market A have a higher time valuation than consumers in market B, so that v A > v B holds in the remainder of the paper. We also assume that downstream firms have constant marginal costs 6 and, following Singh and Vives (1984), that their costs are incorporated through the intercept of the inverse demand function. 7 In the analysis that follows, we study the case where downstream firms set a unit price and cannot discriminate consumers so, in equilibrium, the firm s price equals P i (q i ) v i D(Q), the marginal willingness to pay net of congestion delay costs. Section 6 extends the analysis by relaxing this assumption and studies downstream firms applying first-degree price discrimination. Consequently, for a given input price, w i, the downstream firm i maximizes: π i = q i P i (q i ) v i D(Q) w i, (1) and the first-order condition leads to the following pricing rule: π i q i = 0 P i (q i ) v i D(Q) = w i + q i P i (q i ) + v i D (Q). (2) Eq. (2) shows that the firm s pricing rule has the facility charge (w i ), a traditional monopoly market power markup ( q i P i ) and the marginal congestion cost on firm i s own consumers 6 We assume that congestion does not affect the downstream firms costs, but this could be readily included in our analysis without changing the main results and conclusions. The reason is that congestion does affect firms in that increased congestion raises the full price faced by consumers and therefore final good prices will be lowered by the increased congestion. In the downstream firms profit function, whether congestion raises the costs or reduces the passengers willingness to pay makes no difference. 7 If a i is the inverse demand intercept in market i and c i the marginal cost, we may replace A i by a i c i. 5

6 (q i v i D ). The downstream firm realizes that an additional consumer raises congestion and reduces the price it can charge, but does not internalize the effect on the other firm s consumers. This internalization result was first recognized by Daniel (1995) in the context of airport congestion pricing and explored theoretically by Brueckner (2002). The system of first-order conditions in Eq. (2) for both firms defines the derived demands faced by the input provider q A (w A, w B ) and q B (w A, w B ). The closed form for the derived demands are in Appendix A. Before moving into the supplier s maximization problem, it is useful to compare the downstream price with the welfare maximizing price. In this model, total welfare is: W = qi P i (x)dx v i q i D(Q), (3) i 0 i and the welfare maximizing downstream pricing rule is W q i = 0 P i (q i ) v i D(Q) = q A v A + q B v B D (Q) i {A, B}. (4) A comparison between Eq. (2) and Eq. (4) reveals that, input prices aside, the prices set by downstream firms are not necessarily higher than optimal. If the demand is sufficiently elastic, i.e. the demand-related markup is low compared to the un-internalized externality (e.g. q A P A < q B v B D ), prices will be too low and output too high. This result and its implications for airport pricing have also been discussed in the air transportation literature (see e.g. Pels and Verhoef, 2004). As one of our aims is to analyze the role of congestion on the effects of input price discrimination, we follow much of the literature and assume that demands are linear. This allows us to compare our results to those in the previous literature more transparently, as mainly models with linear demands have been used to study the effect of price discrimination in final goods markets under negative consumption externalities (e.g. Adachi, 2005) and when input buyers participate in multiple markets (e.g. Arya and Mittendorf, 2010). We also assume, for simplicity, that the congestion delay function is linear in the aggregate quantity. Note that assuming linear functional forms does not mean that we confine our analysis to a constant aggregate output, because in presence of within- and cross-market congestion externalities and linear demands, the output effect of price discrimination is not zero when time valuations are different (Czerny and Zhang, 2015). The pricing regimes that we study are uniform pricing, where the facility is restricted to charge all firms the same price per unit of output, and price discrimination, where the facility is allowed to charge different unit prices. 8 We assume throughout the paper that all markets are always served under both pricing regimes. The equilibrium concept that we use is subgame-perfect Nash equilibrium, and we use backward induction to identify it. We first study the case of a profit maximizing facility. 8 There is a distinction between price differentiation and price discrimination in congestible markets (see e.g. van der Weijde, 2014). As in our setting the marginal external cost ( i vi qi D (Q)) is the same for all consumers, there is no difference between discrimination and differentiation, so we use them interchangeably. 6

7 3. Private facility 3.1. Price discrimination When price discrimination is allowed, the facility chooses w A and w B to maximize: Π P D = w A q A (w A, w B ) + w B q B (w A, w B ). (5) where we normalize the input supplier s costs to zero. The first-order conditions lead to the closed-form solutions for w A and w B (see Appendix A) and imply the following pricing rules: w A = 2 q A P A + v A D + q B v B D, (6) w B = 2 q B P B + v B D + q A v A D. (7) Not surprisingly, the input provider also exerts market power and consumers face a double marginalization. In addition, the facility charges the marginal congestion cost that is not internalized by the firm (the last term on the right-hand side of Eqs. (6) and (7)). Therefore, under price discrimination, the final price in each market is higher than the socially optimal price and output is inefficiently low. This result is useful for the welfare analysis below and it is essentially different to the case of final good markets and congestion externalities, where the quantity under price discrimination can be inefficiently high. This is because the downstream firm s markup is not necessarily higher than the marginal external congestion cost, but the sum of the downstream and upstream markups is. In our analysis a crucial aspect is whether w B, the input price in the market with a lower value of time, is higher than w A, the price in the market with higher value of time. In absence of congestion effects and, therefore, of interrelation between markets, the input price is higher in the market with the higher inverse demand intercept (see e.g. Arya and Mittendorf, 2010). This is because with linear demands, the intercept determines the elasticity of the derived demand faced by the input supplier. The market with the higher inverse demand intercept is the less elastic market under uniform prices and, therefore, the market where the discriminating input price will be higher (the strong market). We seek to understand what is the effect of the congestion externality on this. Let A i be the intercept of the inverse demand function for market i. Assuming that the second-order conditions are satisfied, 9 the following lemma summarizes the condition for w B > w A to hold (the proof of this Lemma and all other proofs required in this section are in Appendix A): Lemma 1. The input price under price discrimination is higher in the market with a lower value of time (w B > w A holds) if, and only if, A B > λ 1 = 8 P A P B +5 v A v B D 2 +v B D v B D 4P A P B 6 P B v A D 8 P A P B +5 v A v B D 2 +v 2 A D2 +2 v A D 4P B P A 6 P A v B D, with λ 1 < 1. 9 A sufficient condition is that time valuations are not too distinct in that v B/v A >

8 To understand the intuition behind the Lemma, first consider the case where time valuations are the same in both markets (v A = v B ). In this case, λ 1 = 1 holds and the result obtained in absence of interrelation goes through. When cross congestion effects are symmetric the supplier s incentive to charge a higher price in one market over the other do not change and the input price is higher in the market where the reservation price is higher. Second, consider the case where the reservation price is the same in both markets (A B / = 1), a case where, in absence of congestion and interrelation, it is optimal for the supplier to set a uniform price because the elasticities of the derived demand are the same. If there were only within-market congestion externalities (i.e. absence of interrelation), as in Adachi (2005), it would also be optimal for the facility to set a uniform price. Adachi (2005) shows in final good markets that the price is higher in the market with higher reservation price because it fully determines which market is less elastic when consumption externalities are linear in the quantity. In the case of input markets, this is also the case as it is straightforward to show that the differences in elasticity of the input demand can be fully explained by differences in the reservation price due to the linear demand and congestion assumption. Thus, cross congestion effects drive the incentive to set a higher price in one market over the other when A B / = 1. In our setting, raising the price in one market causes a decrease in congestion costs through decreased demand, which, in turn, causes an increase in the profitability of the other market as the willingness to pay is increased. Consequently, when the reservation price is the same in both markets, it is optimal for the input supplier to set a higher price in the market with low time valuation. Phrased differently, for the input supplier the decreased congestion is more profitable in the market with high time valuations because the increase in willingness to pay is higher. Third, consider the case of different inverse demand intercepts and time valuations, where both effects come into play as Lemma 1 reveals. A lower demand intercept makes the input demand more elastic as it is normally the case with linear demands (in absence of congestion), which gives incentives to decrease the input price, and a lower value of time gives incentives to increase the price in that market because of cross congestion effects. It is straightforward to show that λ 1 decreases as the ratio v B /v A is lower, so that the more asymmetric the congestion effects are, the stronger the incentives to raise the price in market B. This is why λ 1 < 1 and even when the reservation price is larger in market A, the input price can be higher in market B. In the general case, the interplay between the relative size of the inverse demand slopes, the time valuations and the inverse demand slopes determines which market faces a higher input price. Interestingly, it is also straightforward to show that λ 1 > v B /v A holds regardless of the relative size of the inverse demand slopes. Therefore, Lemma 1 implies that it is more likely that the input price is higher in market B if the asymmetry of inverse demand intercepts is lower than the asymmetry of time valuations. If they are similar or A B / is less than the ratio of time valuations, then the input price will be higher in market A. For the congestion effects to overturn the incentives provided by different demand intercepts (elasticities in absence of congestion), the difference between time valuations must be higher 8

9 than the difference between demand intercepts Uniform pricing Under a uniform pricing regime, the profit-maximizing facility maximizes: Π U = w q A (w, w) + q B (w, w), (8) and the first-order condition leads to the following pricing rule (again prices are not informative and are in Appendix A): w = 2 q A P A + v A D + 2 q B P B + v B D (q A + q B ) 2 P B + v B D P A P B + v D P A + v A D P A P B + v D v A D v B D P A P B + v D, (9) where v = (v A + v B )/2 is the average value of time. The pricing rule in Eq. (9) includes a weighted sum of the supplier s markups and a negative term that is also a weighted sum of the marginal congestion cost that is external to each firm. It is straightforward to show that the uniform price in Eq. (9) is not an weighted average of the differentiated prices in Eqs. (6) and (7). What causes this result is the demand interdependency through congestion. As Czerny and Zhang (2015) show in final good markets, the uniform price can be lower than both discriminatory prices in presence of congestion externalities. We obtain a similar result in our setting of input price discrimination, so the uniform price is not necessarily an average of the differentiated prices because of demand interrelation. In the following section we study the relationship between uniform and discriminatory prices in detail The effects of price discrimination on prices and output To study the effect of price discrimination on input prices and output, we use the pricedifference constraint method used by Leontief (1940) and Schmalensee (1981). We assume that the facility maximizes profit subject to the constraint w B w A t. This is, the input supplier cannot differentiate prices more than the exogenous amount t 0. When t = 0, the facility sets the uniform price derived above (Eq. (9)). As t gradually increases, the input supplier is gradually allowed to increase the price differentiation until it reaches a point, t, where it sets the prices w A and w B in Eqs. (6) and (7). The method consists of evaluating the marginal effect of relaxing the constraint on a variable, such as aggregate output. If the sign of the marginal effect does not change in the range 0, t, the overall effect of price discrimination on the variable will have the same sign, as long as the unrestricted input provider sets a higher charge in market B (w B > w A ). If the opposite holds, i.e. w B < w A, the overall effect of price discrimination will have the opposite sign of the marginal effect, because the price discrimination behavior is approached by making t negative. All the derivations needed for the results in this section are in Appendix A. 9

10 For a given value of t 0, t, the facility maximizes: Π = w A q A (w A, w A + t) + (w A + t) q B (w A, w A + t). (10) Totally differentiating the first-order condition Π/ w A, we can obtain the marginal effect on the aggregate output and input prices: dq dw A = v A v B D 2 Ω 1 > 0, (11) = 4 P A 3 v A v B D Ω 2 < 0, (12) dw B = 3 v B v A D 4 P B Ω 2, (13) where Ω 1 and Ω 2 are positive constants. The results that follow from Eqs. (11) (13) are summarized in the following proposition. Proposition 1. When demands are such that the facility sets a higher input price in the market whose consumers have a lower value of time (i.e. w B > w A ), price discrimination: (i) Increases aggregate output. (ii) Decreases the input price in the market where time valuations are higher (A). (iii) Decreases both input prices if time valuations are sufficiently different in that v A 3 v B > 4 P B /D ; otherwise, it increases the input price in the market where time valuations are lower (B). When demands are such that w A > w B holds, the effects are reversed and price discrimination decreases output, increases the input price in market A, and it increases both prices when time valuation are sufficiently different. Our output effect result is an extension of the result in Layson (1998), who shows, in substitute final good markets, that under linear demands the sign of the output effect is determined by the relative magnitude of the gross substitution effect. In our setting outputs are not substitutes nor complements, but the interdependency through congestion generates a similar effect as substitution. An output increase in one market increases the full price of the other market s consumers by means of increased congestion and therefore it induces an output reduction. As the cross effects are proportional to the time valuations (see Appendix A), it is intuitive that the output change is not zero as long as these effects are not symmetric (v A v B ). A difference with Layson (1998) is that the relative magnitude of the cross effects is not the only determinant of the sign of the output effect. That is, if v A > v B holds and the relative magnitude of the cross effects is given, output may rise or fall depending on the relative magnitude of the input prices (the sign of w B w A ). This result on the output effect also extends previous analyses of price discrimination in presence of congestion externalities. Czerny and Zhang (2015) find that price discrimination by a monopoly airline to two classes of passengers (where, as in our case, demands are only interrelated though congestion) always reduces the aggregate quantity under linear 10

11 demands. The key difference lies in the properties of the derived demands, which can differ essentially with the final good demands. Czerny and Zhang (2015) assume that demands are such that the (final good) price is higher in the market where time valuations are higher and this implies that price discrimination cannot increase output. Although this seems adequate in their setting as, for example, one can think of business and leisure passengers, it is not necessarily appropriate for input prices. This is because the assumption that there is a strong market where consumers pay a higher price in equilibrium is not necessarily a good proxy for the relative magnitude of the input prices under price discrimination. It is possible that w B > w A holds and that the equilibrium downstream price is higher in market A. The effect of price discrimination on prices also extends Layson (1998). He shows, under linear demands, that for prices to move in the same direction when price discrimination is allowed, cross-price effects must be asymmetric and the firm s marginal costs must be decreasing. In our setting, only asymmetric cross effects are required (v A v B ) as marginal costs are constant. The difference here is that if the (input) price rises in one market, the aggregate quantity does not necessarily decrease because of congestion effects: when the price in one market increases, the full price in the other market may decrease because of the decreased congestion costs. The results also extend Czerny and Zhang (2015) who find that price discrimination cannot reduce both prices, but only increase them. Again the difference is in their assumption on the relative magnitude of final prices. The intuition of why price discrimination can reduce both prices is similar to the one provided by Layson (1998). In our model, an increase in the input price of one market increases the profitability of the other market, as congestion costs decrease. This increases the consumers willingness to pay and therefore the price that can be charged. Under uniform pricing, the marginal profit of the input provider in each market has a different sign. Consider that the marginal profit is negative for market A under uniform pricing (consistent with w B > w A ). If the marginal profit increases slowly towards zero, the decrease in price towards the optimally differentiated w A will be large. This large decrease may cause a large reduction in the profitability in market B, which was positive at uniform prices, and can make it negative at {w A, w}. This will therefore cause a reduction also in the price in market B. This is what happens when the facility sets a higher input price in the market whose consumers have a lower value of time and time valuations are sufficiently different in that v A 3 v B > 4 P B /D. A similar explanation works for the case where both prices increase. Which of the results in Proposition 1 is more likely to take place depends on the relation between time valuations and reservation prices. In our model, there are three sources of asymmetry between markets: time valuations, reservation prices and inverse demand slopes. All three are arguably correlated through (average) income: a higher income in market A would explain a higher time valuation, and it would also imply that the reservation price is higher and the demand less sensitive to price changes. The sign of the output effect depends on whether w B > w A holds or not. Using Lemma 1, we obtain that output is 11

12 more likely to increase with price discrimination when the ratio A B / is greater than the ratio of time valuations v B /v A and it will decrease if the asymmetry in reservation prices is similar to or higher than the asymmetry of time valuations. This, naturally, will depend on how the differences in income impacts the time valuations and reservation prices and it is ultimately a matter of empirical investigation. Second, the way in that prices change with price discrimination depend also on how asymmetric the time valuations are. Price discrimination is likely to move prices in the opposite direction when the ratio of time valuations v B /v A is not too low (higher than 1/3 is sufficient) and to change prices in the same direction when it is sufficiently low (v B /v A at least lower than 1/3). As explained above, this is because when they are sufficiently different (v A 3 v B > 4 P B /D ), the change in profitability in one market due to the change in the input price of the other is large. The likelihood of v A 3 v B > 4 P B /D is somewhat difficult to assess. One way of casting light into its likelihood is by considering that the differences across markets are caused by differences in trip purpose. Koster et al. (2011), Kouwenhoven et al. (2014) and Shires and De Jong (2009) provide empirical evidence that the ratio of time valuations between business and other users in transport markets is not higher than 3. This suggests that v A 3 v B > 4 P B /D is a rather stringent condition when differences between markets are caused by differences in the proportion of business and other types of travelers. In that case it is more likely that input price discrimination increases the price in one market and decreases the price in the other. Which market faces the high price depends on the relation between time valuations and reservation prices as discussed earlier. If the ratio of demand intercepts is similar as the ratio of time valuations, for example, because income affects reservation prices and time valuations in a similar way, then price discrimination will increase the price in the market with high time valuations (market A). Only when the ratio A B / is greater than the ratio of time valuations v B /v A price discrimination can increase the input price in the low income market (market B) Welfare analysis A full characterization of the marginal welfare effect would be tedious in our case. First, unlike the case of final good markets, under the uniform pricing regime there is, in general, a misallocation of output between markets. This is because downstream firms charge a markup related to demand characteristics and time valuations, so that when the input price is uniform, the marginal willingness to pay is, generally, not the same in each market. To see this, consider the marginal change in total welfare as more discrimination is allowed using the same method as in the previous section: dw = dq A + dq (w A w) q A P A + q A v A D + dq B (w B w) q B P B + q B v B D w q A v A + q B v B D, (14) where the first two terms in square brackets are the final good prices (w i q i P i +q i v i D ) minus the uniform input price input price set by the facility (w), and the third bracketed 12

13 term is the difference between the uniform input price and the marginal external congestion cost. When the input prices are uniform and equal to w, there is still a misallocation effect unless the sum of the demand related markup and the internalized congestion is the same in both markets ( q A P A + q A v A D = q B P B + q B v B D ). Second, as Czerny and Zhang (2015) point out, the presence of congestion externalities gives rise to an effect that works in the opposite direction as the output effect on welfare. Thus, welfare can increase when output is decreased by price discrimination. This section provides a partial characterization of the effect of price discrimination on welfare by deriving sufficient conditions for welfare improvement and deterioration. Rearranging Eq. (14), we get: dw = dq A w A q A P A + q B v B D + dq B w A + t q B P B + q A v A D (15) where the terms in square brackets multiplying the marginal quantity changes are the difference between the input price set by the facility and the socially optimal input price. The welfare analysis can be divided in two cases, namely when price discrimination changes both quantities in the same direction (both either rise or fall) and when price discrimination increases the quantity in one market and it decreases it in the other. We first focus in the latter case. Opposite changes in demand due to price discrimination are a consequence of opposite changes in prices. As discussed in Proposition 1, this happens when time valuations are not too different and, thus, the effect of a price change in one market on the marginal profitability of the other is not large enough to provide incentives to increase or decrease both input prices. In this case, the output increases in the market where the input price decreases and it decreases in the other market. We provide sufficient conditions for welfare improvement when aggregate output increases and for welfare deterioration when aggregate output decreases. As shown in Proposition 1, the sign of the output effect depends on whether w B > w A holds or not, which depends on whether A B / > λ 1 holds or not. First, if demands are such that w B > w A (A B / > λ 1 ), the aggregate output increases and, therefore, the quantity decrease in market B is lower than the increase in market A. As a consequence, from Eq. (15), if the difference in actual and socially optimal input price is positive in market A and higher than in market B for all values of t, then price discrimination necessarily increases welfare. Conversely, if w B < w A holds (A B / < λ 1 ), the aggregate output decreases and the quantity decrease in market A is higher than the increase in market B. Therefore, if the difference in actual and socially optimal input price is always positive and higher in market A, welfare decreases. The conditions for this are summarized in the following proposition: Proposition 2. When time valuations are similar in that v A 3 v B < 4 P B /D, the quantities change in opposite directions with price discrimination and: (i) Price discrimination increases welfare if: λ 1 < A B < λ 2 = 12 P A P B +10 v A v B D 2 +2 v B D v B D 4P A P B 11 P B v A D 12 P A P B +10 v A v B D 2 +2 v A D v A D 4P, B P A 11 P A v B D, 13

14 (ii) Price discrimination decreases welfare if: A B < min λ 1, λ 3, where λ 3 = P A +v A D 4 P B +v A D +5 v B D P B +v B D 4 P A +v B D +5 v A D Let us first discuss part (i), where w B > w A holds and aggregate quantity increases. The reason why welfare increases is that the benefit in the market A from a decreased input price and increased quantity is larger than the loss in market B, where the opposite happens. Therefore, it follows that demand in market B cannot be significantly larger than in market A for this to hold. This is why an upper bound on A B / is needed. It is expected that the market with high time valuations is also the market with low demand price sensitivity, so that v A > v B and P A > P B hold. In this case, it is straightforward to show that λ 2 < 1. In addition, the interval λ 1, λ 2 is non-empty when the ratio of the inverse demand s slopes is less than the ratio of time valuations, i.e. v B /v A < P B / P A. Thus, price discrimination is likely to increase welfare when time valuations are similar (v B /v A > 1/3 is sufficient) and the price sensitivities as well as the reservation prices are more similar. The second part of Proposition 2 is intuitive: if A B / is lower than λ 1, price discrimination increases the price in the market A, which is the market with a higher reservation price and with higher time valuations. This, not surprisingly, is likely to reduce welfare. If P A > P B holds, which is expected to hold if the difference of time valuations across markets is due to differences in income, λ 3 > λ 1 holds and therefore A B / < λ 1 is a sufficient condition for welfare deterioration. λ 3 is only part of the necessary condition in the case where P A < P B and it is sufficient for the welfare loss in the market A to be higher than the gain in B. As a result, if time valuations are similar (a ratio higher than 1/3 is sufficient), the market with high time valuations is also the market with lower demand sensitivity to price changes and the ratio of demand intercepts is similar to or lower than the ratio of time valuations, price discrimination will decrease welfare. The welfare analysis when price discrimination changes both quantities in the same direction is in Appendix A. We choose not to discuss it here because the conditions that make price discrimination to increase or decrease both quantities are rather stringent and not very informative. We show that both prices moving in the same direction is not sufficient for both quantities to move in the same direction because of congestion effects. A change in demand in one market has an impact on the full price of the other market and, if cross congestion effects are not low, this can overturn the effect of the own input price change. The conditions that make quantities to either rise or fall involve an upper and lower bound on the ratio of time valuations and also a restriction on the relationship between time valuations and demand slopes. Moreover, the time valuation in market A has to be more than 5 times larger than in market B, which, as we argue above, seems to be unrealistic in transport markets. Nevertheless, when price discrimination increases the quantity in both markets it increases consumer surplus in both markets and total welfare. The reverse may also happen and price discrimination can decrease both quantities, decrease welfare and consumer surplus. The results of this section show benefits from input price discrimination in the presence 14

15 of negative consumption externalities and that price discrimination can increase consumer surplus. Importantly, the benefits are found in a setting where, in absence of externalities, price discrimination yields lower social welfare. In addition, the conditions for welfare improvement depend strongly on the absolute and relative value of the congestion effects. This suggests that the efficiency of a pricing policy can differ with the level of congestion of the facility even if everything else is invariant (e.g. through different capacity of the facility). A natural expectation when the differences in time valuations arise from differences in income across markets is that the market with high time valuations (A) also exhibits lower demand sensitivity to price changes ( P A > P B ) and a higher reservation price ( > A B ). In this case, it follows from Propositions 1 and 2 that price discrimination is more likely to decrease welfare when the asymmetry of the reservation prices is similar to or higher than the asymmetry in time valuations (i.e. when A B / v B /v A ). This is because under linear demands and congestion the market with a higher demand intercept is the less elastic market. It also follows that for price discrimination to increase welfare, the ratio of reservation prices A B / needs to be higher than the ratio of time valuations. In addition, when time valuations are similar in that v B /v A 1/3 holds, price discrimination is more likely to increase welfare when the ratio A B / is, besides being larger than v B /v A, not too high. For example, when 1/3 < v B /v A < 1/2, price discrimination decreases welfare when A B / < 2/3 and it can increase welfare if 2/3 < A B / < 1. In the following section we analyze price discrimination by a public facility with the aim of comparing the welfare results and shed light on when a broad ban on price discrimination is desirable. 4. Public facility We now study a public facility that maximizes domestic welfare. If the facility were maximizing total welfare, allowing price discrimination would always be optimal and the analysis would be trivial. We introduce a source of divergence from total welfare maximization, namely that consumers and firms may be foreign. Among the many possible domestic-foreign structures we consider the case where market A is fully domestic (passengers and firm A are domestic) and the firm B together with a fraction of the passengers in market B are foreign. The assumption that the market with higher time valuations is the domestic market is, we believe, a realistic assumption if the differences in income across markets are a consequence of differences in trip purpose, as business travel is more frequent in domestic destinations than in international travel. For example, in 2012, the share of business trips was 20%, 30% and 190% higher in domestic destinations than international destinations at London City (LCY), London Heathrow (LHR) and Manchester (MAN) airports respectively (CAA, 2012). In 2011 in Los Angeles International Airport (LAX), the share of business trips was 90% higher in U.S. destinations than in international 15

16 destinations (Unison Consulting, 2011). 10 For simplicity we assume that the fraction of foreign passengers in market B is 1/2 and, therefore, the public facility maximizes the sum of its profit, firm A s profit, the consumer surplus in market A and one half of the consumer surplus in market B: qa W D = P A (x)dx v A q A D(Q) + 1 qb 2 P B (x)dx q B P B (q B ) + w B q B. 0 (16) where the first term in square brackets is total welfare in market A (the sum of the consumer surplus, firm A s profit and airport revenues from market A), the second term in square brackets is the consumer surplus in market B and the third term is the airport s revenue from market B. The incentive to price discriminate is to capture part of the foreign firm s profit and stimulate domestic production. The model can easily be extended to cases where a different share of consumer surplus is taken into account by the facility or to cases where there are foreign passengers in both markets, but results do not change in any significant way. What matters is that there is a clear incentive to reduce the price in one market in detriment of the other, and not so much which is the mechanism that provides this incentive Price discrimination The first-order conditions of maximizing W D with respect to both input prices lead to the input prices w A and w B (see Appendix B for the prices and all derivations of the results in this section). Here, as in the previous section, we present the pricing rules: 0 w A = q A P A + q B v B D, (17) w B = 2 q B 34 P B + v B D + q A v A D. (18) The input price for the domestic firm is a subsidy equal to the downstream markup (q A P A < 0) and the marginal congestion cost that is not internalized by firm A (q B v B D ). 11 This is the first-best pricing rule, as it makes the final price in the market equal to the marginal social cost (see Eq. (4)). The price in the foreign market is the sum of a market power markup and the marginal congestion cost that is not being internalized. The public facility does not subsidize the foreign firm, but the markup is lower than in the private case, as the consumer surplus in this market is partially taken into account because a fraction of the consumers are domestic. 10 Our assumption may be less realistic for air transportation in high income countries with small domestic markets, such as the Netherlands or Switzerland. In those cases, our model may be representative of other transportation markets where congestible facilities provide an input to downstream firms, such as rail transportation. 11 The facility charges the externality imposed on the foreign market because it is profit maximizing to do so (see the pricing rule of the private facility in Eq. (6)) and not because of consumer surplus considerations. 16